How Can Some Math Students Learn So Little in 12 Years of Classes? This afternoon, my Fulbright colleague and math teacher Elizabeth Agro Radday and I were considering this question. Despite a decade of effort, many young students learn little about mathematics except that they hate it. How, we wondered, could twelve years of schooling in math result in such poor outcomes for so many?
The same could be said for history, my discipline, of course. (I have often fretted at how little my students remember about the French Revolution, studied just one year before.) But Liz and I were chatting after visiting the classroom of Pekka Peura, a respected math teacher at the Martinlaakson Lukio, here in Helsinki.
Coincidentally, several years ago Peura came to ask himself the same question Liz and I were kicking around this afternoon. The answers he came up with are really not new, but based on a century’s research about motivation and students’ ownership of their learning.
Peura has implemented a mastery-based, student-controlled math curriculum in his classes. He floats around the room, helping when needed, but allows the students to manage their own learning, to the point of scoring their own final exam.
It isn’t a simple system and must have taken an enormous amount of labor. Peura has uploaded a detailed solution to every problem so his students can see how they are doing as they go–and which skills are the most essential to learn before moving on. When I asked how it worked, Peura pulled out his iPhone and quickly brought up a sample problem. Once the students complete a given “quiz,” they score themselves by using Peura’s rubric. Such a solution is what he showed me on his phone:
Students also have a large card listing each lesson in the course. They score their own work, see where they need more work, and mark down their score on their card. They are constantly asked to reflect on whether they need more help, have a basic grasp on a topic, or fell they have mastered a lesson. They decide when they are ready to tackle a new topic. Grades are based on how far students progress into more complex topics.
Won’t the students cheat and inflate their scores I asked? He smiled and said some might, but he knows where they are in their learning after checking in with them every day. He also inferred that those in charge of their learning begin to take outcomes seriously and understand that moving on to more complex math without mastery of what comes before is a recipe for failure and stress. And students cheat on traditional assessments as well, while often learning less.
Peura even characterized cramming for a test as “cheating” because it is a way of avoiding doing the learning that is really the chief object of study. We teach them to forget as well as to learn, he told me. We make them learn a lot for an exam and then encourage them to forget it so they can cram more information for the next test into their short-term memory. Since this doesn’t lead to real learning but may result in an inflated grade, it is arguably a dishonest or at least inaccurate assessment of their learning. Teachers should find a way to discourage this kind of preparation. One way is to eliminate high-stakes summative assessments.
At lunch, Peura talked about his students’ process of math learning in reference to the Dunning-Kruger Effect. I didn’t know what this was, so he showed me a graph of it on his phone. Many
math students, he argued, at first have an inflated sense of their abilities but are then crushed by their inevitable failure. A teacher’s job is, in part, to move them to the part of the curve at which they can see the relationship between their rising confidence and their growing expertise. Putting students in charge of their own learning, Peura seemed to be arguing, gets them to the right side of the graph more quickly. It makes sense: by offering his students clear and lengthy rubrics that require constant self-reflection and self-assessment, he has removed much if not all of the dark arts from his students’ experience of math.
As a social studies teacher, I wonder how I can make use of such a teaching style. The objective nature of much mathematics makes the self-reflection model fitting. Could it work in teaching the causes and impact of the French Revolution?
“Peura’s Theses”: from Suomen Kuvalehti, “A Teacher Who Does Not Teach,” translated by Google’s miraculous-but-clearly-not-Finnish robot translator, with a few edits by yours truly.
- All students are able to learn when given a suitable pace;
- Teaching characterized by lecture is appropriate for only a small percentage of students. For most it is too slow or two fast;
- Too slow a pace frustrates; too fast a pace prevents proper training for building new data upon knowledge already learned, making it more difficult to absorb; the result is a negative spiral;
- Learning is more effective when the student is required to reflect upon lessons and discusses them with others;
- The teacher’s job is to help students identify their skills and their own way of learning;
- Most traditional assessments are pointless and do not measure actual know-how. Instead, they cause undue stress for many students and decrease their quality of life. Therefore, teachers must work to develop new metrics;
- This method is applicable to mathematics in addition to other subjects as well as primary school teaching.